ABC Tools

Wave Interference, Explained

After reading this you will be able to predict where two ripple sources reinforce and where they cancel, count the fringes behind a double slit, and read the blue-white-red patterns in the ripple tank as solutions of the two-dimensional wave equation.

What the ripple tank shows

Drop two oscillating sources into still water and the surface stops being simple. Where two crests arrive together the water rises higher than either source alone. Where a crest meets a trough the surface stays flat. The result is a fixed pattern of bright and dark bands that does not move even though the water is in constant motion.

The tool steps the wave equation on a 240 by 160 grid. Crests render red, troughs blue, and the still baseline white. With the two-source preset you see a set of curved lines fanning out from the midpoint between the sources. Those curves are hyperbolas, and every point on one of them has the same difference in distance to the two sources. That single geometric fact explains the whole picture.

When to use it, and when not

Use the simulation to build intuition for any linear wave phenomenon: water ripples, sound in air, light through slits. The same math describes all three because all three obey the wave equation to good approximation at small amplitudes. It is a fast way to see why fringe spacing shrinks when you raise the frequency, and why a wider slit separation packs more beams into the same fan.

Do not read it as a literal model of a specific physical tank. The grid is coarse, damping is artificial, and the sponge layer at the edges is a numerical trick, not physics. Nonlinear effects (breaking waves, large amplitudes, wind) are absent entirely. Treat the numbers as a clean version of the ideal case, not a measurement.

Interference needs coherent sources: sources that hold a fixed phase relationship. In the tool every source runs off the same clock, so they stay coherent forever. Real light bulbs do not, which is why Young needed a single slit to feed his two slits in 1801.

The wave equation behind the picture

The height of the surface u(x, y, t) obeys the two-dimensional wave equation:

\frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)

Here u is the surface displacement, t is time, x and y are position, and c is the wave speed on the grid. The right-hand side is the Laplacian \nabla^2 u, which measures how much a point sits below or above the average of its neighbors. Where a point dips below its neighbors, the equation pushes it back up, and the acceleration overshoots. That overshoot is what makes waves oscillate rather than settle.

The tool solves this with a leapfrog scheme. It stores the field at two past times and computes the next one:

u^{n+1}_{i,j} = 2u^{n}_{i,j} - u^{n-1}_{i,j} + \frac{c^2 \Delta t^2}{\Delta x^2}\left( u^{n}_{i+1,j} + u^{n}_{i-1,j} + u^{n}_{i,j+1} + u^{n}_{i,j-1} - 4u^{n}_{i,j} \right)

The superscript n is the time step and the subscripts i,j are grid cell indices. The factor c^2 \Delta t^2 / \Delta x^2 controls stability. In two dimensions the scheme stays stable only while that factor stays below 0.5, the CFL limit. Push it higher and the field blows up to infinity within a few steps.

Where crests reinforce: the path-difference rule

Take two sources a distance d apart, both driven at wavelength \lambda. Pick any point in the tank. Let r_1 and r_2 be the distances from that point to each source. The waves add up constructively when the path difference is a whole number of wavelengths:

r_2 - r_1 = m\lambda, \quad m = 0, \pm 1, \pm 2, \dots

They cancel completely when the path difference is a half-integer:

r_2 - r_1 = \left(m + \tfrac{1}{2}\right)\lambda

Each fixed value of r_2 - r_1 traces a hyperbola with the two sources as foci. The m = 0 line, where the paths are equal, is the straight perpendicular bisector between the sources. That is always a bright fringe. Far from the sources the hyperbolas straighten into rays, so the pattern looks like a fan of beams leaving at fixed angles.

For those far-field rays the angle \theta from the center obeys a simple relation:

d \sin\theta = m\lambda

Raise \lambda (lower the frequency) and the angles spread apart. Raise d and they pack together. That is the whole story of fringe spacing.

Reproducing the two-source preset

The defaults place two sources with a spacing of 60 grid cells and drive them at a wavelength of \lambda = 12 cells (frequency chosen so one full wave spans 12 cells). Count the bright beams and their angles.

  1. The largest usable order satisfies m\lambda \le d, so m \le 60/12 = 5. Orders run from m = -5 to m = 5.
  2. That is 11 bright beams total, including the central m = 0 beam straight ahead.
  3. The first beam sits at \sin\theta = \lambda/d = 12/60 = 0.2, so \theta \approx 11.5^\circ.
  4. The second beam sits at \sin\theta = 24/60 = 0.4, so \theta \approx 23.6^\circ.
  5. The fifth beam sits at \sin\theta = 60/60 = 1.0, so \theta = 90^\circ: it grazes straight sideways and is barely visible.

The angles are not evenly spaced because \sin\theta is linear in m, not \theta itself. The gap from beam 0 to beam 1 is 11.5 degrees, but from beam 4 to beam 5 it is only 37 degrees widening to 90, a much larger jump.

The bright beams crowd near the center and spread wide at high orders, because the angle grows with the arcsine of a linear quantity.

With source spacing d = 60 cells and wavelength \lambda = 12 cells, bright beams leave at 0, 11.5, 23.6, 36.9, 53.1 and 90 degrees. Halving the wavelength to 6 doubles the number of beams and moves the first one down to 5.7 degrees. Doubling the spacing to 120 also doubles the beam count.

The double slit as two sources in disguise

Choose the double-slit preset and a plane wave rolls in from the left. When it reaches the barrier, only the two gaps let it through. Each gap becomes a fresh source of circular waves. This is Huygens' principle: every point on a wavefront acts as a point source. So the double slit is the two-source problem with the slit separation playing the role of d.

Behind the barrier you see the same fan of bright beams. The central beam points straight through. The path-difference rule d\sin\theta = m\lambda gives the beam angles exactly as before. If you widen the slit separation with the slider, watch the beams pack tighter, matching the 1/d scaling.

Slit width matters too. A single wide slit spreads its own diffraction pattern that modulates the brightness of the beams, dimming the outer ones. To see clean, even beams, keep the slits narrow relative to the wavelength.

Reading the colors and the fringes

The colour map is diverging: deep blue for the lowest trough, white for zero displacement, deep red for the highest crest. The contrast slider stretches this scale. High contrast makes small ripples visible but clips the tall ones to flat red or blue.

Bright fringe
A line where the two waves add in phase. The colour swings hard between red and blue as time passes. These are the beams of the fan.
Dark fringe (nodal line)
A line that stays white at all times because the waves cancel. Between every pair of bright beams sits one of these.
Sponge layer
The outer ring of cells where the field is gently attenuated each step. It stops waves bouncing off the wall and re-entering the tank as false interference.

To count fringes, freeze on a moment of peak contrast and count the coloured beams crossing a line drawn parallel to the sources far from them. With the default preset you should count 11, matching the worked example.

Common mistakes

The most common error is pushing the frequency and speed so high that c^2 \Delta t^2 / \Delta x^2 crosses the CFL limit of 0.5. The field then explodes into a checkerboard of maxed-out red and blue within a few frames. If the screen goes to violent static, lower the frequency, not the contrast.

A second mistake is reading fringe spacing near the sources. The clean d\sin\theta = m\lambda rule holds only in the far field, many wavelengths away. Close in, the hyperbolas are strongly curved and the beam picture does not apply yet.

A third is expecting the pattern to drift. Bright and dark fringes are stationary in space as long as the sources stay coherent and fixed. Only the colours pulse. If the fringes wander, you have dropped an extra source with the mouse, adding a third term to the sum.

Related tools

If standing-wave patterns interest you, the Chladni Plate Patterns tool shows the two-dimensional version on a vibrating plate, where sand collects on the nodal lines that stay still. For the wave equation's cousin in field theory, drag magnets in Magnetic Field Lines and watch superposition of fields rather than waves. The Harmonograph composes sine waves along two axes into Lissajous curves, a one-dimensional take on adding oscillations. And if you want to see how the same numerical stepping handles nonlinear motion, the Fluid Flow Simulator integrates the fluid equations where waves and vortices mix.

Frequently asked questions

Why do the fringes not move even though the water does?

The locations where the two waves cancel depend only on the path difference to the sources, which is fixed geometry. The displacement at a bright point oscillates in time, but the point stays bright and its neighbouring dark point stays dark. The pattern is standing, the motion is local.

Why does raising frequency add more fringes?

Higher frequency means shorter wavelength \lambda. The highest order beam satisfies m \le d/\lambda, so shrinking \lambda raises the maximum order and adds beams. Halving \lambda roughly doubles the count.

What is the sponge layer for?

A finite grid has hard walls that reflect waves. Those reflections would come back and interfere with the real pattern. The sponge damps the outer cells so outgoing waves fade before they reach the edge, approximating an open, infinite tank.

Is this how real light makes fringes?

The geometry is identical. Young's 1801 experiment used exactly the d\sin\theta = m\lambda rule to prove light is a wave. The difference is scale: light wavelengths are near 500 nanometres, so the slit separation is measured in fractions of a millimetre rather than grid cells.

Can I trust the exact fringe angles from the tool?

Trust them to within grid resolution. The far-field angles follow d\sin\theta = m\lambda closely, but the 240 by 160 grid quantises both position and wavelength, so expect errors of a degree or two on the outer beams and larger errors very close to the sources.

Try it yourself in the tool

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